A lot of the fussiness of handloading for competition is going to depend upon what kind of competition and how small a group you want to "chase." There are at least 3 or 5 types of competition conducted at 600 yards and all stress varying skills, which is why they are different matches.
Depending upon which type of competitor you talk to and at what level that person competes, you will get varying answers to your questions. As mentioned, many BR, F-class and other serious competitors will consider a weight variance in bullets, brass, powder and primers to be indicative of other defects. This is why they use components geared toward competition, and will choose among them to further refine the quality to the point which they require. As an example, Sierra rightly claims the lion's share of use by serious competitors in service rifle and across the course competitions, possibly the newer 3 gun as well. That being said, almost no BR short range, and few long range shooters serious, about competing, list Sierra as bullet of choice in the NBRSA or IBS equipment lists which accompany match results in the newsletters. These folks are using custom made bullets, made one at a time individually inspected and virtually guaranteed to have no difference in weight, base to ogive length, jacket thickness and concentricity. F-class shooters are beginning to plump for these very expensive bullets, too. There are varying levels of quality in components from there on down. If the groups you are chasing are not normally measured in tenths of an inch, you will not need to explore the wonderful (addictive and expensive) world of premium components.
As for primers, there can be a big difference in downrange results if you are chasing those tenths of an inch. This is why there are favored brands and types of primers listed in those equipment lists. This is also why various super precise primer seaters are sold, including at least one that actually allows you to measure the depth of the primer pocket, the height of the primer cup, and will tell you how many thousandths of an inch you have "crushed" the primer. Again, depending upon your obsession, you may also want to buy things like primer pocket uniformers and flash hole reamers.
Many experienced rifle handloader/competitors/hunters will tell you all this stuff is a waste of time--they are right--from their perspective. There are also folks with vast experience in the BR/long range varmint/f-class world who will tell you it is required to win or even finish in the pack and not behind it.
A friend of mine had some good advice for my son when he started to get serious about rifle accuracy: if you shoot it off a rest listen to Tony Boyer. If you shoot it offhand listen to David Tubb. These days you can read their books too.
"If the enemy is in range, so are you." - Infantry Journal